Making Meaning with Arrays: More Preschooler Division

While my 2 year old napped, my 4 year old and I baked some chocolate chip muffins.

“You know, mama, there are 4 people in all,” she announced, while swirling the mix with a spoon.

“Four people in all?”

“Yes. There used to be 5, when Mamoucha [her grandmother] lived here, but now there are 4.”

“Oh, there are 4 people in our house?”

“Yes.”

I glanced down at the muffin tin. We had made a small batch — only enough to fill 9 baking cups. There were so many mathematical directions our conversation could go.

“Do you think there’s enough muffins for everyone?” I asked her.

She smiled. “Oh, yes, of course!”

I had watched her share different foods before. I wrote about it in a blog post: Everyone Gets A Handful: Preschoolers Explore Division (January 28, 2020). To share out a bag of cheese puffs (pirate booty), she had scooped handfuls. To share out ten strawberries, she hard carefully distributed in rounds, like you would when dealing out playing cards. How would she share these muffins?

…only they weren’t even muffins yet. In front of us lay 9 baking cups filled with gooey muffin batter, arranged in a 3 by 3 array.

“3 for me, 3 for Papa and my brother, and 3 for you!”

She was using the structure of the array to support her thinking!

“Is that fair? That you get 3, but that Papa and your brother have to share 3?”

“Yes, it’s fair. They didn’t help.”

Touché, little one.

When the muffins came out of the oven, she immediately claimed one as her prize. This left 8 muffins, no longer in an array.

“Now there are 8 muffins!” My daughter cackled like the Count on Sesame Street.

“How will we share them with the family now?”

She looked at me, bewildered. With both the pirate booty and the strawberries, she had manipulated the objects to figure out how to share them. With the batter-filled baking cups, she had used the structure of the array. Now there was no apparently mathematical structure to help her. As someone fluent with division, I could see how we could partition out these 8 muffins into little groups of 2, but this comes from mathematical experience. (Dare I say… wisdom.)

She seems stuck, so I encouraged her to move the muffins around. They were fully baked now, with no splash risk.

She moved them around until they fit into a nice 2 x 4 array, going down the left of the muffin tin.

“Oh! Oh! Look!” She pointed to the top row of 2 muffins. “These are for me!” Then she pointed to the second row of 2. “These are for you! Then… these are for Papa, and these are for my brother.”

The structure of the array had once again helped her make sense of this problem.

_____

Later in the week, we baked some chocolate chip cookies.

“How many cookies did we bake?” I asked my daughter.

“1, 2, 3, 4…. 1, 2, 3, 4… that’s 4 for me, 4 for you, 4 for papa, and 4 for my brother!” She exclaimed with delight. “And then we can have a leftover!”

“Who gets the leftover?”

“We’ll figure it out,” she assured me. She probably thought it would be her. My money was on her wiley little brother.

“So that tells me how we can share them,” I continued. “But how many cookies did we bake in all?”

“1, 2, 3, 4… 1, 2… umm…”

She stumbled. Eventually, I rearranged the cookies like this:

She was able to count them with so much more ease, even though there were still elements of the array, like even rows.

Fascinated, I looked up some articles from NCTM. I read one called “Cooking Up Arrays,” by Debra Rawlins, Natasha Hernandez, and William Miller, published in the November/December 2018 edition of Teaching Children Mathematics. 

The authors cite research from Michael Battista and his team about children’s conceptions around arrays.

Battista (1999) and his colleagues (1998) showed that ready-made pictorial grid arrays are difficult for students to understand because they cannot readily recognize the row-by- column structure.

-Debra Rawlins, Natasha Hernandez, William Miller, “Cooking Up Arrays” Teaching Children Mathematics Nov/Dec 2018 (p 144 – 150).

I saw how this rang true for my daughter. In fact, she didn’t think that she could manipulate the objects in the array. They did, however, help her see how she could share or group things, showing that even at age 4 she’s starting to recognize how arrays can support different contexts better than others. (that said, if I had to count by 1s, I think an array would be more helpful and organized than randomly grouped objects, but I do have lots of array experience.)

A study from Taylor Martin, published in 2009 in Child Development Perspectives, offered the following:

Children performed better when they could move the pieces, regardless of whether they were prestructured—they answered correctly more often than when the pieces were already partitioned into the correct groups.

– Taylor Martin, “A Theory of Physically Distributed Learning: How External Environments and Internal States Interact in Mathematics Learning.” Child Development Perspectives no. 3 (December 2009), p 142)

Being able to manipulate and interact with the arrays definitely helped my daughter to play out the question, experiment with some new lines of thinking, and ultimately solve them.

It looks like I’m going to have plenty of time and space to explore these ideas with my own children. Of course, I wonder about the classroom implications. (Read: Dan Meyer’s “But Artichokes Aren’t Pinecones: What Do We Do With Wrong Answers?” from March 10, 2020.) How will this affect how I teach — whether it’s distance learning or when we finally return back to our classrooms? When do students need to mess around with actual materials? How much experience with a mathematical representation does an individual student need in order to leverage the meaning without the objects? How do we ever make time and space for this in our busy classroom lives?

…and what should my daughter and I bake next.